Irregular finite order solutions of complex LDE's in unit disc

Abstract

It is shown that the order and the lower order of growth are equal for all non-trivial solutions of f(k)+Af=0 if and only if the coefficient A is analytic in the unit disc and log+⁡M(r,A)/log⁡(1−r) tends to a finite limit as r→1−. A family of concrete examples is constructed, where the order of solutions remains the same while the lower order may vary on a certain interval depending on the irregular growth of the coefficient. These coefficients emerge as the logarithm of their modulus approximates smooth radial subharmonic functions of prescribed irregular growth on a sufficiently large subset of the unit disc. A result describing the phenomenon behind these highly non-trivial examples is also established. En route to results of general nature, a new sharp logarithmic derivative estimate involving the lower order of growth is discovered. In addition to these estimates, arguments used are based, in particular, on the Wiman-Valiron theory adapted for the lower order, and on a good understanding of the right-derivative of the logarithm of the maximum modulus.